Walker Breakdown with a Twist: Dynamics of Multilayer Domain Walls and Skyrmions Driven by Spin-Orbit Torque

Current-induced dynamics of twisted domain walls and skyrmions in ferromagnetic perpendicularly magnetized multilayers is studied through three-dimensional micromagnetic simulations and analytical modeling. It is shown that such systems generally exhibit a Walker-breakdown-like phenomenon in the presence of a current-induced dampinglike spin-orbit torque. Above a critical current threshold, corresponding to typical velocities of the order of tens of m/s, domain walls in some layers start to precess with frequencies in the gigahertz regime, which leads to oscillatory motion and a significant drop in mobility. This phenomenon originates from complex stray field interactions and occurs for a wide range of multilayer materials and structures that include at least three ferromagnetic layers and finite Dzyaloshinskii-Moriya interaction. An analytical model is developed to describe the precessional dynamics in multilayers with surface-volume stray field interactions, yielding qualitative agreement with micromagnetic simulations.

Figure 1

Time evolution of current-driven isolated twisted domain walls. (a),(b) Micromagnetic snapshots that depict precessional motion for $j=4.5\times {10}^{11}{\mathrm{A/m}}^2$ (just above the critical current). (a) The $x$-$z$ cross section and (b) schematic side view of transient layers. Time $t=0$ corresponds to the static profile and the precession initiates in layer $i=13$ (arrows depict the direction of DW rotation). In the layer below it ($i=12$), the DW oscillates (without precession), while in all other layers, the DW profile remains unchanged (although quick readjustments of ${\psi }_i$ are still present as discussed at the end of section V). Flat arrows show schematically the direction of the Thiele effective force $F_i$ from the dampinglike spin-orbit torque acting on several layers. (c) Average DW velocity as a function of current density, with empty (filled) lines or dots indicating the presence (absence) of full $2\pi$ precession and the solid line corresponding to the steady-state DW solution [42]. (d) Precessional frequency as a function of current density and layer number with dashed lines representing guides ($f\sim \sqrt{j^2-j_{\mathrm{cr}}^2}$) to the eye. Interfacial DMI is fixed to $D=1.0{\mathrm{mJ/m}}^2$. The shading in panels (c) and (d) indicates which layers, if any, are precessing, as labeled in panel (c).

Figure 2

The evolution of twisted skyrmions right above the critical current. (a),(b) $x$-$z$ cross sections of low- and high-DMI skyrmions and (c),(d) in-plane micromagnetic profiles of the precessing layer (arrows depict the direction of spin rotation). The upper (lower) row corresponds to $D=0.25{\mathrm{mJ/m}}^2$, $j=12.5\times {10}^{11}{\mathrm{A/m}}^2$, $i=9$ ($D=2.0{\mathrm{mJ/m}}^2$, $j=17.0\times {10}^{11}{\mathrm{A/m}}^2$, $i=14$). The corresponding static DW width (${\mathrm{\Delta }}_i$) and DW angle (${\psi }_i$) are depicted as a function of DMI in Fig. 1 of Ref. [42].

Figure 3

The dynamic properties of twisted skyrmions. (a) Skyrmion velocity as a function of current density. Continuous lines correspond to analytical DW model calculations and symbols correspond to 3D micromagnetic simulations for skyrmions, with closed symbols indicating precession-free motion and open symbols denoting precessional motion. (b) Average topological charge as a function of layer number and current density for $D=1.0{\mathrm{mJ/m}}^2$. The topological charge is computed as $N=1/4\pi \int \mathbf{m}\cdot \left(\mathrm{\partial }\mathbf{m}/\mathrm{\partial }x\times \mathrm{\partial }\mathbf{m}/\mathrm{\partial }y\right)\mathrm{d}x\mathrm{d}y$ [4]. (c) Instantaneous topological charge as a function of time for $D=1.0{\mathrm{mJ/m}}^2$ and $j=12\times {10}^{11}{\mathrm{A/m}}^2$.

Figure 4

The dynamics of twisted DWs given by analytical theory. (a) DW velocity and precessional frequency as a function of current density. (b) DW position as a function of time and current density in the proximity of the critical current ($j_{\mathrm{cr}}=6.4\times {10}^{11}{\mathrm{A/m}}^2$). (c) The same as in (b) but after subtracting the linear $⟨v⟩\ast t$ contribution, where $⟨v⟩$ is the time-averaged velocity and $t$ is the time. For all cases, $D=1.0{\mathrm{mJ/m}}^2$.

Figure 5

Critical parameters as a function of DMI constant $D$ and number of multilayer repeats $\mathcal{N}$. (a) Critical layer number $i_{\mathrm{cr}}$ for the precession onset and (b) the Bloch layer number $i_{\mathrm{Bloch}}$ [42]. (c) Critical current $j_{\mathrm{cr}}$ for the onset of the first Walker breakdown event and (d) the corresponding critical velocity $v_{\mathrm{cr}}$. Film parameters are for a film with $f=1/6$, $Q=1.4$, $\alpha =0.3$. Brown color in Figs. (a),(c) separates regions for which the critical current (if exists) exceeds $j=50\times {10}^{11}{\mathrm{A/m}}^2$, in Fig. (d) the brown color indicates regions with $v_{\mathrm{cr}}$ higher than $50\mathrm{m/s}$, and in Fig. (b) it indicates regions with no Bloch layers.

Figure 6

(a) Critical current $j_{\mathrm{cr}}$ and (b) critical velocity $v_{\mathrm{cr}}$ as a function of DMI for $\alpha =0.3$. (c) Critical current $j_{\mathrm{cr}}$ and (d) critical velocity $v_{\mathrm{cr}}$ for $D=1.0{\mathrm{mJ/m}}^2$ as a function of $\alpha$. Results in panels (b)–(d) are shown for several values of the quality factor Q and scaling factor f. Continuous lines represent the results obtained from the analytical model, and discrete symbols represent the results of 3D micromagnetic simulations of straight DWs (circles) and skyrmions (stars). Dotted lines in Fig. (b) represent $v_{\mathrm{cr}}$ given by our simplified model [Eq. (9), (10)]. The color shading in panel (a) indicates the map of skyrmion eccentricity. In all figures, $\mathcal{N}=15$.

Figure 7

The effect of current on twisted DWs and skyrmions. (a) Average DW and skyrmion velocity, where continuous (dotted-dashed) lines represent the data given by our proposed analytical model (low-current twisted skyrmion theory [42]) and points (stars) represent the data given by micromagnetic simulations for multilayer DWs (skyrmions). (b) The position of DW as a function of layer number and current density for $D=1.0{\mathrm{mJ/m}}^2$. (c) The separation between the top and bottom DWs and skyrmions as a function of current density and DMI. For skyrmions, the decoupling data correspond to the offsets of their bottom and top centers (the interlayer radius variation is within $6\mathrm{nm}$). Magnetic parameters are $\mathcal{N}=15$, $f=1/6$, $Q=1.4$, ${\theta }_{\mathrm{SH}}=0.1$, $\alpha =0.3$.

Figure 8

Numerical solution of dynamic Eqs. (B21)–(B23) for the SOT-driven motion. Films have (a) $\mathcal{N}=1$, (b) $\mathcal{N}=2$, (c) $\mathcal{N}=3$ multilayer repeats. All the cases depict a steady-state dynamics that can also be identically described by Eqs. (C1) and (C2), with the exception of panel (c), which also depicts the precession and oscillation of DWs for $j>j_{\mathrm{cr}}=1.03\times {10}^{13}{\mathrm{A/m}}^2$. Material and external parameters are $D=2.0{\mathrm{mJ/m}}^2$ (at $j=0$, all DWs are Néel with ${\psi }_i=+\pi /2$), ${\theta }_{\mathrm{S}H}=+0.1$, $\alpha =0.3$, $f=1/6$, $Q=1.4$, $H_z=0$.

Figure 9

Numerical solution of Eqs. (C21)–(C23): (a),(b) for $D=0.25{\mathrm{mJ/m}}^2$ and (c),(d) for $D=2.0{\mathrm{mJ/m}}^2$. For all values of $C_3$, the expression $\sin ({\psi }_0)+\sin ({\psi }_1)+\sin ({\psi }_2)\approx 0$ [Eq. (C20)] cannot be satisfied. Note that the domains of functions ${\psi }_i(C_3)$ are constrained from above by $C_3$, so the range is plotted in accordance with it. $\mathcal{N}=3$, $Q=1.4$, $f=1/6$, $\alpha =0.3$.